Scilab Function
Last update : 6/7/2004

eval_cshep2d - bidimensional cubic shepard interpolation evaluation

### Calling Sequence

  `[zp [,dzpdx, dzpdy [,d2zpdxx,d2zpdxy,d2zpdyy]]] = eval_cshep2d(xp, yp, tl_coef)`

### Parameters

- `xp, yp`: two real vectors (or matrices) of the same size
- `tl_coef`: a tlist scilab structure (of type cshep2d) defining a cubic Shepard interpolation function (named `S` in the following)
- `zp`: vector (or matrix) of the same size than `xp` and `yp`, evaluation of the interpolant `S` at these points
- `dzpdx,dzpdy`: vectors (or matrices) of the same size than `xp` and `yp`, evaluation of the first derivatives of `S` at these points
- `d2zpdxx,d2zpdxy,d2zpdyy`: vectors (or matrices) of the same size than `xp` and `yp`, evaluation of the second derivatives of `S` at these points

### Description

This is the evaluation routine for cubic Shepard interpolation function computed with `cshep2d`, that is :

          zp(i) = S(xp(i),yp(i))
          dzpdx(i) = dS/dx(xp(i),yp(i))
          dzpdy(i) = dS/dy(xp(i),yp(i))
          d2zpdxx(i) = d2S/dx2(xp(i),yp(i))
          d2zpdxy(i) = d2S/dxdy(xp(i),yp(i))
          d2zpdyy(i) = d2S/dy2(xp(i),yp(i))

### Remark

The interpolant S is C2 (twice continuously differentiable) but is also extended by zero for (x,y) far enough the interpolation points. This leads to a discontinuity in a region far outside the interpolation points, and so, is not cumbersome in practice (in a general manner, evaluation outside interpolation points (i.e. extrapolation) leads to very inacurate results).

### Examples

    // see example section of cshep2d

    // this example shows the behavior far from the interpolation points ...
    deff("z=f(x,y)","z = 1+ 50*(x.*(1-x).*y.*(1-y)).^2")
    x = linspace(0,1,10);
    [X,Y] = ndgrid(x,x);
    X = X(:); Y = Y(:); Z = f(X,Y);
    S = cshep2d([X Y Z]);
    // evaluation inside and outside the square [0,1]x[0,1]
    m = 40;
    xx = linspace(-1.5,0.5,m);
    [xp,yp] = ndgrid(xx,xx);
    zp = eval_cshep2d(xp,yp,S);
    // compute facet (to draw one color for extrapolation region
    // and another one for the interpolation region)
    [xf,yf,zf] = genfac3d(xx,xx,zp);
    color = 2*ones(1,size(zf,2));
    // indices corresponding to facet in the interpolation region
    ind=find( mean(xf,"r")>0 & mean(xf,"r")<1 & mean(yf,"r")>0 & mean(yf,"r")<1 );
    color(ind)=3;
    xbasc();
    plot3d(xf,yf,list(zf,color), flag=[2 6 4])
    legends(["extrapolation region","interpolation region"],[2 3],1)
    xselect()

### See Also

`cshep2d`,

### Authors

  Robert J. Renka
  B. Pincon (scilab interface)